CALCULATION OF MICROSTRESSES IN TEXTURED POLYCRYSTALS WITH CUBIC CRYSTAL SYMMETRY

The problem of the analytic determination of microstresses in textured polycrystals with 
cubic symmetry under general static loading has been solved. The solution is based on the 
expansion of macroscopic and microscopic stress fields into hydrostatic and deviatoric 
portions. This essentially simplifies the description of microstresses in polycrystals.


INTRODUCTION
Under rejection of the hypothesis of a homogeneous medium, macro- stresses in the textured polycrystal are determined as the average of microstresses in grains of a polycrystal over a representative volume of the medium containing a sufficiently large amount of the grains.Microstresses are determined by the applied macrostresses and the dis- tinction between the effective and local values ofthe elastic constants.The well-known models of Sachs and Reuss (Taylor and Voigt) are based on the assumption of homogeneity of stresses (strains) in the sample.Consequently these models do not consider the influence of surrounding grains and texture on the stresses in the grains.The estimation of the influence oftexture and interactions between grains on the microstresses under various orientations of external load is the topic of this paper.
Polycrystals with cubic crystal symmetry are considered.For clear description ofthe interaction between grains and the influence oftexture on the microscopic stresses, microstresses are found in the grains ofnon- textured material and a polycrystal with special texture.The obtained results are compared with the data for a homogeneous material (i.e. a single crystal).The solution is based on the known effective values of elastic characteristics of the polycrystal.For a quasi-isotropic system, this is the Aleksandrov's solution (Aleksandrov, 1965).For a textured material, these are exact elastic constants of the two-component texture (001)[100] + (001) [110]  (Mityushov and Berestova, 1995).

DETERMINATION OF MICROSTRESSES IN GRAINS OF TEXTURED POLYCRYSTALS
Local stresses in a non-textured (quasi-isotropic) material are found using Eshelby's solution (Eshelby, 1957) for the deformation of an elastic spherical grain of cubic symmetry embedded in an infinite homogeneous matrix.Let e be the constrained strain of a bigger size grain from material of the matrix with elastic characteristics c*.Equate the stress in this grain under the given uniform strain of the polycrystal (e) to the stress in the cubic symmetry grain with properties c under the same uniform strain (Shermergor, 1977)   (2.1)Here e T is the tensor of incompatibility of strains.
Based on Eshelby's solution, the tensor of the constrained strain and the tensor of incompatibility of strains are related to each other in the following manner: ee=Ne w or er=We e (2.2) The components of the Eshelby's tensor N W -1 are given by Nijkl---*pqkl L OX(j)OXq "-OX(i)OXq J (2.3) where Gpi(X,X') is the Green's tensor of the infinite homogeneous medium; v0 is the volume occupied by the grain.

DETERMINATION OF MICROSTRESSES IN
GRAINS OF TEXTURED POLYCRYSTAL Determination of microstresses in a textured polycrystal with cubic crystal symmetry has been realised for the example of a two-component texture (001)[100] + (001) [110], where the components are taken in equal concentrations.The polycrystal is a macro-transverse-isotropic system, i.e. the system has a plane of symmetry.
For the two-component medium, the compliance tensor at an arbi- trary point of the system can be written as S /(S1 $2) "+-$2, (3.1) where S1 and S 2 are the compliance tensors for the orientations [100] and [110], respectively, and , is a random indicator function that indicates which texture component is present.() is equal to the volume content of the first component, and in the equal volume case () 1/2.
Further, for determining the average stress over a volume which is occupied by the separate orientation, we use the Hooke's law for an arbitrary point of the system.The condition (3.1) allows us to write the Hooke's law in the form e s2r + (Sl s2)r.
This considers, in particular, the influence ofthe inhomogeneity ofthe medium on the change of the tangential stresses 7"(001)[100] and 7"(110)[i01 versus the application of external tensile load in the (001) plane of the fixed crystallite.The relations (3.6) in the case can be written in the form c7 sin(2b) (or), (3.10) where (a) is the external tensile load, b is the angle between the tensile axis and the [001] direction.
4. SUMMARY The ratios of the shear stresses in the planes (100), ( 110) and the tensile load versus the angle between the tensile axis and the direction [001] are shown in Fig. for gold, lead and aluminium for the mon0crystal, the quasi-isotropic polycrystal and the textured two-component polycrystal.
From these figures, it can be concluded that inhomogeneity can have a significant impact on the distribution of microstresses in grains of a polycrystal.The texture of the material and the anisotropy of the single crystal elastic properties had the most impact on the values of the local stresses.The largest inhomogeneity of stresses, which is characterised by the differences of the critical shear microstress in the inhomogeneous and homogeneous material (single crystal) are shown by materials with a high indicator of elastic anisotropy (Pb, Au).The largest differences for the (001)[110] system were achieved when the tensile axis and the [100] direction coincided, while for the (001)[100] system they were achieved when the tensile axis and the [110] direction coincided.
The obtained solution allows the estimation of the inhomogeneity of the shear stresses, as distinct from the well-known models of Sachs and Reuss.Stresses in the considered crystallographic systems ofthe inhomo- geneous textured material are twice that of the corresponding homo- geneous material.Consequently, use of the indicated models in this case yields a very rough approximation.Dependence of the ratio of the tangential stresses r for the two crystallographic systems to the tensile stress (tr) on the direction of its application for lead (a,b), gold (c,d) and aluminium (e,f) in textured, quasi-isotropic and .homogeneousmaterial.denotes the curve obtained for textured polycrystal (001)[100]+(001)[110]; FIGUREDependence of the ratio of the tangential stresses r for the two crystal- The Eshelby's tensor is expanded into members in a similar manner N IN E1N + E2N + E3N 3N1E1 + 2N2E2 + 2N3E3.(2.16)In this case independent coefficients ofexpanded tensors are determined by bulk modulus and Poisson's coefficient u* of the matrix material,